Photography requires light. When there is not enough light, a camera either has to use a flash or reduce the shutter speed. In the case of your phone, it reduces the shutter speed. This would happen with a camera too if you do not use a flash. The only ways around not using flash and still getting a blur-free image is to raise the camera's digital sensor sensitivity, and/or use a wide aperture lens. Small cameras have small sensors, so when you raise the sensitivity, the photos become very "noisy," or grainy looking. Most small cameras do not have very wide aperture lenses either.
So...if you want to take photos indoors, get more lights in your house, or use the flash on the camera, or buy a larger camera with a bigger sensor and wider aperture lens.
I like the colors of the photos from my iPhone 5 but since I have a child too many photos are blured because she moves. What would you suggest me to buy of a fast focusing camera and not too big. Thank you. Bjarne